me 203 staticsme203.cankaya.edu.tr/uploads/files/me203_wk11_ch08.pdf · me 203 statics...

ME 203 Statics

Asst.Prof.Dr.Turgut AKYÜREK !Çankaya University, Ankara

Friction

T.Akyürek ME 203 Statics – Friction

In-Class Activities:

• Check Homework, if any

• Reading Quiz

• Applications

• Analysis of a Wedge

• Analysis of a Belt

• Concept Quiz

• Group Problem Solving

• Attention Quiz

Today’s Objectives:

Students will be able to:

a) Determine the forces on a wedge.

b) Determine tension in a belt.

Wedges And Frictional Forces On Flat Belts

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W1. A wedge allows a ______ force P to lift a _________ weight W.

A) (large, large) B) (small, large)

C) (small, small) D) (large, small)

2. Considering friction forces and the indicated motion of the belt, how are belt tensions T1 and T2 related?

A) T1 > T2 B) T1 = T2

C) T1 < T2 D) T1 = T2 eµ

Reading Quiz

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How can we determine the force required to pull the wedge out?

When there are no applied forces on the wedge, will it stay in place (i.e., be self-locking) or will it come out on its own? Under what physical conditions will it come out?

Wedges are used to adjust the elevation or provide stability for heavy objects such as this large steel pipe.

Applications

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How can we decide if the belts will function properly, i.e., without slipping or breaking?

Belt drives are commonly used for transmitting the torque developed by a motor to a wheel attached to a pump, fan or blower.

Applications

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How can you determine the tension in the cable pulling on the band?

Also from a design perspective, how are the belt tension, the applied force P and the torque M, related?

In the design of a band brake, it is essential to analyze the frictional forces acting on the band (which acts like a belt).

Applications

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WEDGES

• The wedge is a simple machine which may be used to transform horizontal input forces into vertical output forces. – The angle of friction may be used to simplify the

solution.

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It is easier to start with a FBD of the wedge since you know the direction of its motion.

Note that: a) the friction forces are always in the direction opposite to the motion, or impending motion, of the wedge; b) the friction forces are along the contacting surfaces; and, c) the normal forces are perpendicular to the contacting surfaces.

WA wedge is a simple machine in which a small force P is used to lift a large weight W.

To determine the force required to push the wedge in or out, it is necessary to draw FBDs of the wedge and the object on top of it.

Analysis of a Wedge

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To determine the unknowns, we must apply EofE, ∑ Fx = 0 and ∑ Fy = 0, to the wedge and the object as well as the impending motion frictional equation, F = µS N.

Next, a FBD of the object on top of the wedge is drawn. Please note that: a) at the contacting surfaces between the wedge and the object the forces are equal in magnitude and opposite in direction to those on the wedge; and, b) all other forces acting on the object should be shown.

Analysis of a Wedge

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Now of the two FBDs, which one should we start analyzing first?

We should start analyzing the FBD in which the number of unknowns are less than or equal to the number of E-of-E and frictional equations.

Analysis of a Wedge

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NOTE: If the object is to be lowered, then the wedge needs to be pulled out. If the value of the force P needed to remove the wedge is positive, then the wedge is self-locking, i.e., it will not come out on its own.

Analysis of a Wedge

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SPECIAL APPLICATIONS Belt Friction

Consider a flat belt passing over a fixedrough cylinder, as shown here.

The belt has impending motion as shown.

The relationship between the tensions TL (larger tension) and TS (smaller tension) at the ends of the belt is given by:

…where e is the base of the natural logarithm & the Greek letterβ(beta) is the angle of contact between the belt and cylinder in degrees.

e!µ"


SIDEBAR - Flat Belts


Detailed analysis (please refer to your textbook) shows that T2 = T1 e µ β

where µ is the coefficient of static friction between the belt and the surface. Be sure to use radians when using this formula!!

If the belt slips or is just about to slip, then T2 must be larger than T1 and the motion resisting friction forces. Hence, T2 must be greater than T1.

Consider a flat belt passing over a fixed curved surface with the total angle of contact equal to β radians.

Belt Analysis

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Consider a flat belt passing over a fixed curved surface with the total angle of contact equal to β radians.

Belt Analysis

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Belt Analysis

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Belt Analysis

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1. Draw a FBD of the crate. Why do the crate first?

2. Draw a FBD of the wedge.

3. Apply the E-of-E to the crate.

4. Apply the E-of-E to wedge.

Given: The crate weighs 300 lb and µS at all contacting surfaces is 0.3. Assume the wedges have negligible weight.

Find: The smallest force P needed to pull out the wedge.

Plan:

Example

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+ ∑ FX = NB – 0.3NC = 0

↑+ ∑ FY = NC – 300 + 0.3 NB = 0

Solving the above two equations, we get NB = 82.57 lb = 82.6 lb, NC=275.3 lb = 275 lb

The FBDs of crate and wedge are shown in the figures. Applying the E-of-E to the crate, we get

NB

300 lbFB=0.3NB

FC=0.3NCNC

15º

15º

FC=0.3NC

NC

FD=0.3ND

ND

P

FBD of Crate

FBD of Wedge

Example

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NB = 82.6 lb

300 lb

FB=0.3NB

FC=0.3NCNC = 275 lb

15º

15º

FC=0.3NC

NC

FD=0.3ND

ND

P

FBD of Crate

FBD of Wedge

Applying the E-of-E to the wedge, we get ↑+ ∑ FY = ND cos 15° + 0.3 ND sin 15° – 275.2= 0;

ND = 263.7 lb = 264 lb

→+ ∑ FX = 0.3(263.7) + 0.3(263.7)cos 15° – 0.3(263.7)cos 15° – P = 0;

P = 90.7 lb

Example

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2. The boy (hanging) in the picture weighs 100 lb and the woman weighs 150 lb. The coefficient of static friction between her shoes and the ground is 0.6. The boy will ______ ?

A) Be lifted up B) Slide down C) Not be lifted up D) Not slide down

1. Determine the direction of the friction force on object B at the contact point between A and B.

A) B) ←

C) → D)

Concept Quiz

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Given: Blocks A and B weigh 50 lb and 30 lb, respectively.

Find: The smallest weight of cylinder D which will cause the loss of static equilibrium.

Plan:

Group Problem Solving

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Plan:

1. Consider two cases: a) both blocks slide together, and, b) block B slides over the block A.

2. For each case, draw a FBD of the block(s).

3. For each case, apply the E-of-E to find the force needed to cause sliding.

4. Choose the smaller P value from the two cases.

5. Use belt friction theory to find the weight of block D.


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Case a (both blocks sliding together):

↑ + ∑ FY = N – 80 = 0

N = 80 lb

+ ∑ FX = 0.4 (80) – P = 0

P = 32 lb

PB

A

N

F=0.4 N

30 lb

50 lb


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Case b has the lowest P (case a was 32 lb) and thus will occur first. Next, using a frictional force analysis of belt, we get

WD = P e µ β = 5.812 e 0.5 ( π/2 ) = 12.7 lb

A Block D weighing 12.7 lb will cause the block B to slide over the block A.

↑ + ∑ Fy = N cos 20° + 0.6 N sin 20° – 30 = 0 N = 26.20 lb

→ + ∑ Fx = – P + 0.6 ( 26.2 ) cos 20° – 26.2 sin 20° = 0 P = 5.812 lb

20º

30 lb

0.6 NP

N

Case b (block B slides over A):


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2. In the analysis of frictional forces on a flat belt, T2 = T1 e µ β. In this equation, β equals ______ .

A) Angle of contact in degrees B) Angle of contact in radians

C) Coefficient of static friction D) Coefficient of kinetic friction

1. When determining the force P needed to lift the block of weight W, it is easier to draw a FBD of ______ first.

A) The wedge B) The block

C) The horizontal ground D) The vertical wall

W

Attention Quiz

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SQUARE-THREADED SCREWS: SCREW JACKS

• The screw jack hasa threaded stem that turns in a fixed base. – As a force P is applied

to the handle, the stem turns and moves up (or down) lifting (lowering) the load W.

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T.Akyürek ME 203 Statics – Friction !28

Frictional Forces on Screws

tab


• Inclination of the plane is given by:

…where p is the pitch of the thread and d the mean diameter of thethreads.

The screw can be analyzed by considering the screw thread as an inclined plane rolled on a cylinder.

#s


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The friction force Fm and normal force NR have been replaced by their resulting R, which acts atangle (angle of static friction) from the normal NR.

Shown is the force diagram for the freebody diagram raising the loadof a small isolated screw element.

#s

#s


• From the equations of equilibrium:

#s

Eliminating R between (a) and (b) gives:

#s

#s

#s

(a)

(b)

#s

#s


• To produce motion, the moment of the force P on the handle of the screw jack must be equal to the moment of the force Fx at the mean radius d/2 of the screw—both about the axis of the screw.

Thus:


P is the force applied to the handle.L the length of the handle. W the load to be lifted by the jack screw.θ the inclination of the plane of the screw given by:and φ the angle of friction.

• Substituting for Fx from

…where:…gives:

#s

#s

#s

SQUARE-THREADED SCREWS: SCREW JACKS• To lower the load, the friction force is reversed:

#s

– The weight will remain in place, even if M is removed.

• If φ θ, then M is a positive number, and the screw is said to be self-locking.

>#s

• If φ θ, then M is a negative number, and the weight will lower by itself, unless M continuesto be applied.

<#s

AXLE FRICTION: JOURNAL BEARINGS

• Journal bearings are used to provide lateral support to rotating machines like shafts & axles. – If the bearing is partially lubricated or not lubricated

at all, the methods of this chapter may be applied.


• The journal bearing shown supports a shaftthat is rotating at a constant speed.

To maintain the rotation, a torque or moment couple M must be applied.The load W and momentcouple M cause the shaftto touch the bearing at A.The shaft climbs up on the bearing until the forces and couple are in equilibrium.


• The shaft touching the bearing at A gives rise to reaction NR and friction force Fk as shown in the free-body diagram.

#s


• For equilibrium:

#k

#k

For small angles of friction:

…thus:

#k #k $k

$k


SIDEBAR - Wood Bearings

SPECIAL APPLICATIONS Disk Friction, Thrust Bearings

Consider a hollow rotating shaft whose end is in contact with a fixed surface, as shown.

Assume the force between the rotating shaft and fixed surfaceis distributed uniformly.

External torque M required to cause slipping to occur is given by:

…where P is the thrust force,Ro the outside radius, and Ri the inside radius of the hollow shaft.

$k

SPECIAL APPLICATIONS Disk Friction, Thrust Bearings

$k

Consider a hollow rotating shaft whose end is in contact with a fixed surface, as shown.

The largest torque, without slipping & transmitted by a disk clutchcan be found from these equations, where fk has been replaced by fs.

For a solid shaft of radius R:

$k $s

$k


Rolling Resistance

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ROLLING RESISTANCE

• If a wheel is moved without slipping over a horizontal surface while supporting a load,a force is required to maintain uniform motion. – Thus, some kind of rolling resistance must be present.

ROLLING RESISTANCE

• The usual method for describing rolling resistance is based on deformation of the surface as shown:

As h is nearly equal to r :

Let resultant reaction between the surface and rollers act at point A.

From equilibrium:

ROLLING RESISTANCE

• The usual method for describing rolling resistance is based on deformation of the surface as shown:

The distance a is called the coefficient of rolling resistance. The values of a are given in millimeters or inches. They vary substantially for the same material with different values of W and r.


Rolling Resistance

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Example 8.11

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Homework 4 (Part 3)

! Problem 8.62 ! Problem 8.75 ! Problem 8.102

Due date: 1st hour of Week 12 (for both Part 1 & Part 2 & Part 3)

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English – Turkish Dictionary

Frictional force Sürtünme kuvveti impending Eli kulağında coefficient Katsayı, sabite

tangent teğet Area of contact Temas alanı crush ezmek

slipping kayma apparent Aşikar, belli, görülür

crate Kasa, sandık

tipping Yana yatma dump boşaltma Vending machine Satış otomatı

ladder merdiven wedge kama Square-threaded Kare-kesit dişli

screw vida Self-locking Kendiliğinden kilitlenen

turnbuckle fırdöndü

Flat-belt Düz kayış V-belt V-kayışı cord Şerit, kordon

drum tambur Pivot-bearing Muylu yatak Collar-bearing Bilezikli yatak

Journal-bearing Kaymalı yatak Sanding machine Taşlama makinesi lubricate yağlama

Rolling resistant Yuvarlanma direnci pulley makara winding sarma

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